I'm pretty sure Emma frost in Origins doesn't work with the Emma Frost in First Class
Well, to even see her called "Emma Frost" you have to watch the trailers, because in the movie she isn't called that. I'm sure she was intended to be Emma Frost originally, but I think the "official" story is now that she was just Kayla's sister Emma who was not Emma Frost. This was said by Lauren Shuler Donner IIRC, and it implies that they see FC and Origins as in some sense being in the same continuity if they felt the need to clarify that at all.
Origins is constantly ignored/contradicted in FC, it stands to reason that Origins is pretty much non canon for the X-Men movies now
For whatever reason, FC simply wasn't very careful in maintaining consistency with the previous films, including X3 and the first film. By the above logic that means those films are also "pretty much non canon". We know, however, that both
The Wolverine and DOFP are intended to be in continuity with X3, despite the fact that FC appeared to contradict it.